Functions: Domain

Understand what the domain of a function is, learn about real-world input constraints (like radius and tax brackets), and see how domains are represented on graphs.

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Learning Guide: The Coordinate System

Learning Guide: Domain of a Function

The domain of a function is the set of all possible input values (usually xx-values) for which the function is defined and produces a valid real number.

Common Domain Restrictions

Domain constraints can arise from physical reality, defined rules, or mathematical operations that are not defined over the real numbers:
  • Circle Area (Physical Constraints): The area of a circle as a function of its radius is A(r)=πr2A(r) = \pi r^2. Since a physical radius cannot be negative, the domain is restricted to r≥0r \ge 0.
  • Tax Brackets (Piecewise Constraints): In income tax systems, tax rates are defined using brackets (piecewise intervals). For example: a 10% tax rate for income up to 5,0005,000 (0≤x≤50000 \le x \le 5000), and a 15% tax rate for income above 5,0005,000 (x>5000x > 5000). Each bracket represents a part of the domain.
  • Rational Functions (Division by Zero): For the function f(x)=1x−3f(x) = \frac{1}{x - 3}, division by zero is undefined. Since the denominator cannot be 00, we set x−3≠0x - 3 \neq 0, which gives a domain of all real numbers except x=3x = 3.
  • Radical Functions (Square Roots): For the function g(x)=x−2g(x) = \sqrt{x - 2}, we cannot take the square root of a negative number in real mathematics. The expression under the square root must be non-negative, so x−2≥0x - 2 \ge 0, which gives a domain of x≥2x \ge 2.

Domains on Graphs & Boundaries

A function's domain can be determined by looking at the horizontal extent of its graph. If the domain starts or ends at a specific boundary, the way we draw the endpoint depends on whether the boundary is included:
  • Open Circle (small empty circle): Used for strict inequalities (>> or <<) where the boundary point itself is not included. For example, if the domain is x>5x > 5, the point at x=5x = 5 is drawn as a small open circle.
  • Solid Point (normal filled point): Used when the boundary point is included (≥\ge or ≤\le).
xy(1, 0)(2, 0)Open circle (Excluded)Solid point (Included)g(x) = x - 1Domain: x > 1f(x) = √(x - 2)Domain: x ≥ 2
Figure 2: Comparison of endpoint styles on a graph. An open circle excludes the endpoint, while a solid point includes it.
Learning Topics

Frequently Asked Questions

Why does the x-coordinate always come first in an ordered pair?
By mathematical convention, coordinates are always written in alphabetical order as (x,y)(x, y). This standardized order ensures that anyone around the world can communicate and locate points on a coordinate plane consistently without ambiguity.
What happens if we input a value outside the domain?
If you input a value outside the domain, the function is undefined for that value. For example, in f(x)=1xf(x) = \frac{1}{x}, inputting x=0x = 0 results in division by zero, which has no defined mathematical value.
How can you identify the domain of a function from its graph?
To find the domain from a graph, look at the horizontal span along the xx-axis. Find the leftmost and rightmost points of the graph, taking note of whether the endpoints are solid (included) or open circles (excluded).